Chapter 5
Section 5.1: Introduction to Ratios
Ratio: the quotient of two quantities. A ratio has the same properties as a
fraction. Using x and y as variables, a
ratio can be denoted as a fraction
(x/y ) or in
colon notation (x : y). Each term mean the same thing and one term
and be used instead of the other.
Usually if a fraction is used for the ratio, the colon notation is used
like 81/5 : 1 instead of
fraction notation. Most generally, if
the fraction notation is used, the fraction is simplified. An example of this is the ratio of 1.5 to
6. 1.5/6 is not a
proper fraction, but multiplying both the top and bottom by 2 creates 3/12,
and through further simplification or dividing both the numerator and the
denominator by 3 I yield 1/4 which is a proper fraction
and 1.5/6 = 1/4.
Section 5.2: Rates and Unit Prices
A rate is the ratio of 2
different kinds of measure. An example
of this would be suppose after 200 miles I filled my gas tank up with 20
gallons of gas. The ratio would be 200
miles/20 gallons. By
reducing the fraction I get 10 miles/1 gallon or 10 miles
per gallon (mpg).
The ratio of the price to the
number of units is called the unit price or
unit ratio. An example of
this is suppose you bought 12 boxes of cereal at $36.00. The unit price would be $36/12
boxes of cereal and by simplifying I obtain $3.00 per box of cereal.
Section 5.3: Proportions
Two fractions are called a
proportion. When one fraction of a
proportion is a multiple of the other they are then proportional.
Let a, b, c, d be variables
where a/b = c/d. The quickest way to find out if the variables
are proportional is to cross multiply where a ∙ d = b ∙ c. An example of this would be 4/13
= 12/39. By cross
multiplication 4 ∙ 39 = 156 and 13 ∙ 12 = 156 implies that 4
∙ 39 = 13 ∙ 12. Because the
terms are equal, the fractions are proportional. Now consider 8/19 = 4/9. 8 ∙ 9 = 72 and
19 ∙ 4 = 76 because 72
≠ 76, the entries are not proportional.
Consider the proportion x/a
= b/c we need to get x all by itself, and there are two
ways to solve this equation. The first
way is to multiply both sides of the equation by a, then x = ab/c. The other way is to cross multiply which
creates the equation c ∙ x = a ∙ b.
Dividing both sides by c I obtain x = ab/c which
is the same as the above equation. An
example of this is the proportion x/2 = 4/8. This equation implies 8 ∙ x = 2 ∙
4 which implies
x = 8/8
= 1 so the proportion is proportional if x = 1.
Another form of a
proportional equation to be solved is a/x = b/c. By cross multiplying this equation, then x
∙ b = a ∙ c. To get x
isolated divide both sides of the equation by b which yields x = a ∙
c/b. An example of this
would be 3/x = 7/21. By cross multiplying 7 ∙ x = 21 ∙
3 which implies 7 ∙ x = 63. By
dividing both sides of the equation by 7 then x = 9 so to keep the proportion
proportional then x = 9.
Section 5.4 Applications of
Proportions
See pages 334 to 338 of Basic
Mathematics 9th Edition by Marvin L. Bittinger
Section 5.5 Geometric
Applications
Two or more triangles are
considered similar if all of the legs are proportional by the same number. 
From the above two figures it
is easy to see that the bigger triangle is 3 times as big as the smaller one
because each leg on the bigger triangle is 3 times the size of the
corresponding leg in the smaller triangle.
Consider the figures below of
similar triangles
Because the triangles are
similar all of the legs are proportional.
In solving the variables x & y, the variable term = a known
term. Using the above example and we
solve for x first. Then x/4 = 9/3
which implies x/4 = 3
implying x = 12 to keep the triangles similar. Likewise y/5
= 9/3 implies
y/5 = 3 implying y =
15. Looking at the pattern, each leg of
the bigger triangle is 3 times as big as the corresponding leg of the smaller
triangle.
Likewise if two or more
figures are the same, but one is larger than the rest then each leg of the
figures is proportional.
Chapter 6
Section 6.1 Percent Notation
p% is p/100 or p per 100. p% can also be represented
by p ∙ 0.01 or p ∙ 1/100. An example would be showing 73% is :
in Decimal Notation, 73% = 73
∙ 0.01 = 0.73
in Fraction Notation, 73% = 73
∙ 1/100 or 73/100
in Ratio Notation, 73% = 73/100
= 73 : 100.
Let p% be a percentage, to
convert this into decimal notation, replace
(%) with (∙ 0.01) or
move the decimal point two places to the left.
If we convert 240% into fractional form 240% = 240/100
= 240/100 = 22/5. In ratio form it is 240 :
100, or it can be reduced to 24/10 = 12/5
or 12 : 5. In decimal notation 240% =
240 ∙ 0.01 using the formula shown above = 2.40. By moving the decimal two positions to the
left, 2 4 0. =
2.4 0 = 240%.
á_â
Either of the above method
work to achieve the same answer.
Likewise to convert a decimal noted number to a percentage, either
multiply by 100 or move the decimal point two spaces to the right. Follow either method by a %. For example let us look at 2.875. Multiplying it by 100 is
2.875
∙ 100 = 287.5 by adding a % to the end 2.875 = 287.5%. If we move the
decimal point two spaces to the right we get 2.8 7 5 is 2 8 7.5%.
â_á
We get the same answer either
way.
Section 6.2 Percent to
Fraction Notation
To convert
a fraction to a percentage; first the fraction into decimal notation. Then convert
it from decimal notation to a percentage the way demonstrated above. Let us look at 3/12, we
need to simplify it before continuing 3/12 = 1/4. First we put the fraction into decimal form, 1/4
= 0.25. Now converting it from decimal
notation to a percentage, multiply by 100 which leaves us with
_
25%. Now let us consider 4/9,
converting it to decimal form we obtain 0.4.
_
By converting it to a
percentage we obtain 44.4%. Since most
percentages do not have a repeating decimal, unless otherwise instructed, round
it to the nearest decimal value of what you’re dealing with. For instance the above value to the nearest
hundredth would be 44.44%.
Section 6.3 Solving Percent
Problems Using Percent Equations
Some
equivalent meanings of words in story problems.
“of” means
multiply (“ ∙ ” or “ X ”)
“what” means the
use of any letter
“is” means
equals (=)
“%” means ∙ 0.01 or ∙ 1/100
remember that ∙ 100 can have a “ % ” after
Examples:
Problem: What is 30% of 50
2 2 2222
x = 30∙.01∙ 50
x = .30 ∙ 50
x = 15 is the solution
15 is what percent of 45
22 2 2 2 2
15 =
a ∙ 0.01 ∙
45 dividing by 45
1/3 =
a ∙ 0.01
_
.3 = a ∙ 0.01
331/3% = a or the solution is 331/3%
amount = percent number ∙ base where the base is the highest amount or
100%
Examples
What is 52% of 110?
â â ââ æ æ
a = 52 ∙0.01 ∙ 110
a = 52 ∙ 1.10
a = 57.2 is the solution
“what
number” also means “what”
24 is 72% of what number
â â ââ æ â
24 = 72 ∙0.01 ∙ x
24 = 0.72 ∙ x dividing both sides by 0.72 I obtain
331/3 = x where x is the solution or the base
“what
percent” means “what” expressed in percent notation
22 is what percent of 78
â â â â â
22 =
p ∙ 78
11/39 = p
because p must be in percent notation,
p = (11/39 ∙ 100) %
p = 288/39% is the solution
*NOTE* In a test
environment I will not expect you to give the exact fractional percentage as
above unless specifically asked for. The
nearest whole number is good enough when work is shown.
Section 6.4 Solving Percent
Problems Using Proportions
The way of solving a is with a method like:
Number → P ─ a ← Amount
100 → 100 ─ b
← Base
Translate and solve: 124% of what is 72?
By using the above formula, 124/100
= 72/b then solve for b, b = 7200/124
which implies that b = 1800/31 = 582/31
The rest are done like in the
book, pages 387 to 389 of Basic Mathematics 9th Edition by Marvin L. Bittinger.
For section 6.5, read pages
392 to 398 of Basic Mathematics 9th Edition by Marvin L. Bittinger.
Section 6.6
Sales Tax, Commission, and Discount.
sales tax = purchase price ∙ sales tax rate
total price = purchase price + sales tax
by substitution we obtain
total price = purchase price + purchase price ∙ sales
tax rate
= purchase
price ∙ ( 1 + sales tax rate )
Example:
The tax rate here in Boone is 6%. What is the total price of a
$1835.69 furnace?
First we familiarize and translate: The sales tax
is 6% = 0.06; the purchase price is 1835.69; and let the total price be T. We assemble the equation like:
T = 1835.69 ∙ ( 1 + 0.06 ) = 1835.69 ∙ 1.06
We then solve the equation:
T = 1835.69 ∙ 1.06 =
1945.8314
To check the answer, we solve it by
using the first total price equation
above:
Let S be the sales tax, then
S = 1835.69 ∙ 0.06 = 110.1414.
We then calculate T = 1835.69
+ 110.1414 = 1945.8314.
Since both T’s equal each
other, we can determine that T = 1945.8314.
We now state the answer in proper
terms:
Since the cents can be only
represented in the tenths and hundredths of the answer, we round the answer to
the nearest hundredths would imply that the total price is $1945.83. For further examples of how to do the 5 step
process, look at the examples of Basic Mathematics 9th Edition by Marvin L. Bittinger.
commission = commission rate ∙ sales
For an example what is the
commission given when the sales are $34,000.12 and the commission rate is 43%?
Let C be the commission, then
C = .43 ∙ $34,000.12 = $14,620.05
Discount = original price
∙ rate of discount
= original
price – ( original price ∙ rate of discount )
= original
price ∙ ( 1 – rate of discount )
For examples of this, look at
page 409 of Basic Mathematics 9th Edition by Marvin L. Bittinger.
Section 6.7 Simple and
Compound Interest
For the simple and
compound interest rate formula’s let I be the interest,
t is the amount of time in years, P is the
principal, and r is the interest rate.
The simple interest rate
formula is:
I = P ∙ r ∙ t
For the compound
interest rate and using the above variables, let A be the amount, and n be the
amount of times the interest is computed per year.
The compound interest formula
is:
A = P ∙ ( I + r/n
)n ∙ t
For the examples of this,
look at pages 414 to 417 of Basic Mathematics 9th Edition by Marvin L. Bittinger.
Section 6.8 Interest on
Credit Cards and Loans
For Credit
Cards and Loans
For a minimum monthly
payment, the minimum percentage is usually given
For determining the interest
per each time period of compiling the interest, a
simple
interest formula for determining the interest is used
For determining the interest
for more than 1 time period, the complex
formula
is used.
For a
Amortization table, if interest is compounded more than once before a payment,
the complex interest formula is used, otherwise the simple interest formula is
used.
For examples of calculating
rates on any of these, see pages 421 to 426 of
Basic Mathematics 9th Edition
by Marvin L. Bittinger.